Preheat the oven to 375°F (190°C). Grease a baking dish with butter or non-stick spray.
In a large mixing bowl, whisk together the eggs until the yolks and whites are fully combined and the mixture is slightly frothy.
Add the milk, salt, and black pepper to the eggs. Continue whisking until the mixture is well integrated and slightly thickened.
Stir in the shredded cheese and chopped vegetables (if using) until evenly distributed.
Pour the combined mixture into the prepared baking dish, spreading evenly. Place in the oven and bake for 25-30 minutes, or until the top is golden brown and the casserole is set in the center when gently shaken.
Remove the casserole from the oven once the edges are puffed and golden, and a knife inserted into the center comes out clean. Let it rest for 5 minutes before slicing and serving.